Stress-Strain Theory
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Stress-Strain Theory. Under action of applied forces, solid bodies undergo deformation, i.e., they change shape and volume. The static mechanics of this deformations forms the theory of elasticity, and dynamic mechanics forms elastodynamic theory. . u(x+dx). dx. dx’. dx. dx’. u(x). x. x’.

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Stress-Strain Theory

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Stress strain theory

Stress-Strain Theory

Under action of applied forces, solid bodies undergo deformation, i.e., they change shape and volume. The static mechanics of this deformations forms the theory of elasticity, and dynamic mechanics forms elastodynamic theory.


Stress strain theory

u(x+dx)

dx

dx’

dx

dx’

u(x)

x

x’

Length squared: dl = dx + dx + dx = dx dx

2

2

2

2

i

i

1

2

3

dl = dx’ dx’ = (du +dx )

2

2

i

i

i

i

= du du + dx dx + 2 dudx

i

i

i

i

i

i

Strain Tensor

After deformation

Displacement vector: u(x) = x’- x


Stress strain theory

u(x+dx)

dx

dx’

dx

dx’

u(x)

x

x’

Length squared: dl = dx + dx + dx = dx dx

2

2

2

2

i

i

1

2

3

dl = dx’ dx’ = (du +dx )

2

2

i

i

i

i

Length change:dl - dl = du du + 2du dx

2

2

= du du + dx dx + 2dudx

i

i

i

i

i

i

i

i

i

i

du = du dx

Substitute

i

i

j

dx

j

Strain Tensor

After deformation

(1)

into equation (1)


Stress strain theory

u(x+dx)

dx

dx’

dx

dx’

u(x)

x

x’

(1)

into equation (1)

Length change:dl - dl = U U

2

2

Strain Tensor

Length change:dl - dl = du du + 2du dx

i

i

2

2

i

i

i

i

(du + du + du du )dx dx

du = du dx

=

i

j

j

k

k

i

Substitute

dx

dx

dx

dx

i

i

j

dx

j

j

i

i

j

Strain Tensor

After deformation

(2)


Stress strain theory

1 light year

Problem

V > C


Stress strain theory

1 light year

Problem

V > C

V < C

Elastic Strain Theory

Elastodynamics


Stress strain theory

dL L’-L

e

=

=

=

Length Change

L L

xx

Length

L’

L’

L

L

Acoustics


Stress strain theory

dL L’-L

e

=

=

=

Length Change

L L

xx

Length

L’

L’

L

L

Acoustics


Stress strain theory

dL L’-L

e

=

=

=

Length Change

L L

xx

Length

L’

L’

L

L

Acoustics

No Shear Resistance = No Shear Strength


Stress strain theory

Tensional

dw

du

dz

Acoustics

dw, du << dx, dz

dx


Stress strain theory

dxdz+dxdw+dzdu-dxdz

AreaChange

(dz+dw)(dx+du)-dxdz

=

=

+ O(dudw)

dx dz

dx dz

Area

dw du

=

+

dz

dx

e

+

dw

=

du

xx

U

=

dz

e

zz

Acoustics

really small

big +small

big +small

Infinitrsimal strain

assumption: e<.00001

Dilitation

dx


Stress strain theory

k

e

-

P =

(

)

+

Bulk Modulus

xx

U

=

e

dx

zz

1D Hooke’s Law

pressure

strain

-k

du

Infinitrsimal strain

assumption: e<.00001

F/A =

Pressure is F/A of outside

media acting on face of box


Stress strain theory

k

F/A =

(

)

+

xx

Compressional

Source or Sink

k

Larger = Stiffer Rock

=

e

e

zz

zz

Hooke’s Law

Dilation

e

k

U

Infinitrsimal strain

assumption: e<.00001

e

k

-

P =

(

)

+ S

+

xx

Bulk Modulus


Stress strain theory

Newton’s Law

..

..

-

-

dP

dP

r

r

;

w =

u =

dx

dz

Net force = [P(x,+dx,z,t)-P(x,z,t)]dz

density

x,

..

k

Larger = Stiffer Rock

r

u

P (x,z,t)

P (x+dx,z,t)

ma = F

-dxdz


Stress strain theory

Newton’s Law

1st-Order Acoustic Wave Equation

..

P

-

r

u =

u=(u,v,w)

..

..

-

-

dP

dP

r

r

;

w =

u =

dx

dz

density

k

Larger = Stiffer Rock

P (x,z,t)

P (x+dx,z,t)


Stress strain theory

Newton’s Law

1st-Order Acoustic Wave Equation

..

P

-

r

u =

(1)

(3)

(4)

..

..

k

P

= -

U

(2)

..

]

P

-

[

u =

1

r

..

k

]

P

-

[

P =

1

r

(Newton’s Law)

(Hooke’s Law)

Divide (1) by density and take Divergence:

Take double time deriv. of (2) & substitute (2) into (3)


Stress strain theory

Newton’s Law

2nd-Order Acoustic Wave Equation

..

k

]

P

-

[

P =

1

r

..

k

P

P =

r

k

c =

2

Substitute velocity

r

..

2

P

c

P =

2

Constant density assumption


Stress strain theory

Summary

..

..

P

-

k

r

k

]

u =

P

-

[

1. Hooke’s Law: P

P =

= -

U

1

r

2. Newton’s Law:

3. Acoustic Wave Eqn:

k

c =

2

r

..

2

P

c

;

P =

2

Constant density assumption

Body Force Term

+ F


Stress strain theory

Problems

1. Utah and California movingE-W apart at 1cm/year.

Calculate strain rate, where distance is 3000 km. Is it e or e ?

2. LA. coast andSacremento moving N-S apart at 10cm/year.

Calculate strain rate, where distance is 2000 km. Is is e or e ?

xx

xx

xy

xy

3. A plane wave soln to W.E. is u= cos

(kx-wt) i.

Compute divergence. Does the volume change

as a function of time? Draw state of deformation boxes

Along path


Stress strain theory

U

U

n

dl

U

= lim

k

e

-

A

P =

(

)

+

A 0

xx

+ U(x,z+dz)cos(90)dx

+ U(x,z+dz)cos(90)dx

- U(x,z)dz

dxdz

dxdz

dxdz

dxdz

(x+dx,z+dz)

n

n

e

zz

Divergence

= U(x+dx,z)dz

>> 0

= 0

No sources/sinks inside box.

What goes in must come out

Sources/sinks inside box.

What goes in might not come out

U(x,z)

U(x+dx,z)

(x,z)


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